Connecting forward and backward autoregressive models

The aim of this paper is to describe procedures for computing the matrix polynomial defining a vector backward autoregressive recursion from the matrix polynomial defining a vector forward autoregressive recursion. Direct procedures for computing the backward polynomial which do not involve a solution of a matrix Lyapunov equation are described. A novel interpretation is also included of a known procedure which involves the computation of covariance data via the matrix Lyapunov equation. This procedure depends on a standard result connecting forward and reverse time state-space models. A comparison involving operation counts is given of the algorithms.